Substituting these into the formula for a ^{y, z), we get 



^..(y^ 2) = V,G,(y, z) + MyG2(y, z) + M^GgCy, z) 



where G^, Gj, and G3 are known linear homogenous functions of F^, F^, and F3. By substi- 

 tuting this expression for a (y, z) into the above definitions of V„, M , and M , we obtain: 



o r XX ^'^''' x'y' z' 



V^=V^/GidA + My/'G2dA + M^yCadA 



My = V^ /zGjdA + My ACsdA + M^ /zGjdA 

 M^ =-V^ /yGidA - My A'GjdA - M^ ryG3dA 



Then inserting the following conditions (one at a time) 



V = 1; 



X ' 



V, =0; 



V =0; 



X ' y 



It is seen that the G functions must satisfy the following relations: 



/CjdA = 1; J GjdA = 0; J GgdA = 

 /zGjdA = 0; f zG^dA = 1; fzQ^AX = <d 

 /yGjdA = 0; fyG^dA = 0; f yG^dX = 1 



Then the terms in W, depending upon a , become: 



_| 1 r rV,Gj(y, z) + M G2(y, z) + M^G3(y, z)]2 

 W I = / - -. — dA 



My =0 



Mz = 



My = 1; 



M^ = 



M = 



M^ = 1 



XX terms 



Hence 



^^. aw r G? r Gj 



— — = — = V^ J dA + M^ J —L 



dx dV area E area r- 



— = V / dA + M - 



X -^ P y J 



ax 



aM. 



aw 

 aM 



dA + M. 



dA + M. 



j G,G3 



area E 



G2G3 



dA 



/ 



E 



dA 



f G3G1 r G3G2 f 



J —TdA + M J -— dA + M, J 



2l 

 E 



dA 



These equations are three of the elastic equations for the beam. The other three can be 

 derived from the a and o terms of the strain energy and will give expressions for 



47 



